Understand Solving Equations and Inequalities - 6th Grade Math

A teacher asks: "Is $x=9$ a solution to $2x=18$ ?" Choose the statement that correctly checks by substitution.
Answer: Substitute: $2(9)=18$ , so $18=18$ (true). Yes, $x=9$ is a solution. Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For the equation 2x=18 testing x=9, the solving process involves substituting to see if it satisfies. Substitution: pick x=9, replace in 2x=18 to get 2(9)=18, evaluate to 18=18, which is true, so x=9 is a solution. In contrast, an inequality like x>4 might have multiple solutions, but this equation has x=9 as the solution. For example, if testing x=8:2(8)=16≠18 false, but here x=9 works as 18=18 true. The correct statement is identified by proper substitution, showing 2(9)=18 so 18=18 true. A common error is mis-substituting, like saying 2(9)=18 but then claiming 9=18 false, or arithmetic errors.
Sarah claims that $x = 3$ is a solution to the inequality $5x - 7 ≤ 2x + 2$ because when she substitutes, she gets $8 ≤ 8$ , which is true. However, when she solves the inequality algebraically, she gets $x ≤ 3$ . Which statement best explains this situation?
Answer: Sarah's substitution confirms that $x = 3$ is indeed a solution, and it's the boundary value Sarah's work is completely correct. When $x = 3$ : $5(3) - 7 = 15 - 7 = 8$ and $2(3) + 2 = 8$ , so $8 ≤ 8$ is true. Solving algebraically: $5x - 7 ≤ 2x + 2$ , so $3x ≤ 9$ , thus $x ≤ 3$ . The value $x = 3$ is the boundary value where equality holds. Choice A is wrong because both methods agree. Choice C is wrong because $5(3) - 7 = 8$ . Choice D is wrong because inequalities can include boundary values.
Test each value in the set $\{1,2,3,4,5,6\}$ to find all solutions to the inequality $x>3$ .
Answer: $\{4,5,6\}$ This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality x > 3 and the test set {1,2,3,4,5,6}, where you determine which values satisfy it by substituting each one. For example, for x>3 from the set, test each: 4>3 true, 5>3 true, 6>3 true, but 1>3 false, 2>3 false, 3>3 false (since > excludes equal), so solutions are {4,5,6}; inequalities can have multiple solutions, unlike equations which typically have one. For instance, checking fully: 1>3 false, 2>3 false, 3>3 false, 4>3 true, 5>3 true, 6>3 true, confirming {4,5,6}. The correct solution is identified by substitution as {4,5,6}. A common error is including the boundary incorrectly, like adding 3 when > excludes it, or not testing all values and omitting some true ones. To test systematically: (1) list the set {1,2,3,4,5,6}, (2) substitute each into x>3 and check, (3) mark true values like 4,5,6, (4) collect solutions as {4,5,6}; remember the inequality direction, as x>3 excludes 3 but includes greater values.
Compare the two equations using the test value $x=4$ . Equation 1: $x+2=6$ Equation 2: $2x=6$ Which statement is true?
Answer: $x=4$ is a solution to Equation 1 only. Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For testing x=4 in Equation 1: x+2=6 and Equation 2: 2x=6, the process involves substituting into each. Substitution: for Eq1, 4+2=6=6 true; for Eq2, 2(4)=8=6 false. In contrast, if it were an inequality, multiple might work, but here it's equations. So x=4 is a solution to Equation 1 only. The correct statement is identified by separate substitutions. A common error is miscalculating, like 2(4)=6, or confusing the equations.
From the set $\{1,2,3,4,5\}$ , which values make the inequality $x>3$ true? (Check each value.)
Answer: $\{4,5\}$ This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality x>3 and the test set {1,2,3,4,5}, where you determine which values satisfy it by substitution: pick a value like x=4, replace (4>3), evaluate and check if true—yes, so x=4 is a solution; inequalities like x>3 can have multiple solutions ({4,5} here), while equations like x+5=12 typically have one (x=7). For example, with inequality x>3 and set {1,2,3,4,5}, check x=1:1>3 false, x=2:2>3 false, x=3:3>3 false, x=4:4>3 true, x=5:5>3 true, so solutions are {4,5}. Correct solution identification comes from substituting each value and confirming if the inequality holds. Common errors include including the boundary like x=3 when > excludes equality, not testing all values, or arithmetic errors. Testing systematically: (1) list the set {1,2,3,4,5}, (2) test each like x=3:3>3 false, (3) mark true values like x=4 and x=5, (4) collect solutions {4,5}. Remember the inequality direction: > means strictly greater, so 3 is not included.
Which values from the set $\{6,7,8,9\}$ make the inequality $x\le 7$ true? (Check each value.)
Answer: $\{6,7\}$ This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality x ≤ 7 and the test set {6,7,8,9}, where you determine which values satisfy it by substituting each one. For example, x=6: 6≤7 true, x=7:7≤7 true, but x=8:8≤7 false; inequalities can have multiple solutions, including the boundary for ≤. For instance, checking: 6≤7 true, 7≤7 true, 8≤7 false, 9≤7 false, so {6,7}. The correct solution is identified by substitution as {6,7}. A common error is excluding the boundary, like omitting 7 when ≤ includes equal, or not testing all. To test systematically: (1) list {6,7,8,9}, (2) check each against ≤7, (3) mark true like 6 and 7, (4) collect {6,7}; remember ≤ includes equality.
For which value from the set { $4, 5, 6, 7$ } does the equation $2(x - 3) = x + 1$ become false when you substitute it?
Answer: Values $x = 4, 5, 6$ all make the equation false when properly substituted Substituting each value: For $x = 4$ : $2(4-3) = 2(1) = 2$ and $4 + 1 = 5$ , so $2 = 5$ (false). For $x = 5$ : $2(5-3) = 4$ and $5 + 1 = 6$ , so $4 = 6$ (false). For $x = 6$ : $2(6-3) = 6$ and $6 + 1 = 7$ , so $6 = 7$ (false). For $x = 7$ : $2(7-3) = 8$ and $7 + 1 = 8$ , so $8 = 8$ (true). Only $x = 7$ makes the equation true, so $x = 4, 5, 6$ all make it false.
From the set $\{0,1,2,3,4,5\}$ , which values make the inequality $2x<10$ true? (Test each value.)
Answer: $\{0,1,2,3,4\}$ Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For the inequality 2x<10 with the set {0,1,2,3,4,5}, the solving process involves determining which values satisfy it by substituting each one. Substitution: for x=4, 2(4)=8<10 true; for x=5, 2(5)=10<10 false since 10 is not less than 10. In contrast, an equation like x+5=12 has typically one solution. For this inequality, test: 2(0)=0<10 true, 2(1)=2<10 true, 2(2)=4<10 true, 2(3)=6<10 true, 2(4)=8<10 true, 2(5)=10<10 false, so solutions {0,1,2,3,4}. The correct solution set is identified by substitution, excluding the boundary where it's equal. A common error is including x=5 by mistake, thinking 10<10 is true, or arithmetic errors in multiplication.
A student claims $x=8$ is a solution to the equation $x-3=6$ . Is $x=8$ a solution? (Substitute to check.)
Answer: No, because $8-3=5$ , not 6 This question tests understanding of solving equations as finding which values make the statement true, using substitution to check the claimed value. The solving process involves the given equation x-3=6 and testing x=8 by substitution: replace the variable (8-3=5), evaluate (5=6?), check if true—no, since 5≠6, so x=8 is not a solution; for inequalities, test similarly, but equations require exact equality. For example, with equation x-3=6, substituting x=8 gives 8-3=5, and 5≠6, so it's false; if it were x-3<6, then 5<6 would be true, but here it's equality. Correct solution identification comes from accurate substitution and comparison to verify if both sides match. Common errors include arithmetic mistakes like 8-3=6 or 8-3=11, or confusing with inequality. Testing systematically: (1) take the value x=8, (2) substitute into x-3=6 to get 5=6, (3) check equality: false, (4) conclude it's not a solution. Always double-check subtraction and equality.
Test the values in $\{0,2,4,6,8\}$ to find all solutions to the inequality $x<6$ .
Answer: $\{0,2,4\}$ This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality x < 6 and the test set {0,2,4,6,8}, where you determine which values satisfy it by substituting each one. For example, x=4:4<6 true, x=0:0<6 true, but x=6:6<6 false (not less); inequalities can have multiple, excluding boundary for <. For instance, checking: 0<6 true, 2<6 true, 4<6 true, 6<6 false, 8<6 false, so {0,2,4}. The correct solution is identified by substitution as {0,2,4}. A common error is including boundary like 6 when < excludes equal, or arithmetic oversight. To test systematically: (1) list {0,2,4,6,8}, (2) check each against <6, (3) mark true up to 4, (4) collect {0,2,4}; no solution if none, but here there are.
Consider the equation $2x - 3 = x + 1$ . If you solve this algebraically, you get $x = 4$ . Now test whether each value from the set { $2, 3, 4, 5$ } actually satisfies the original equation by substitution. What do you discover?
Answer: Only $x = 4$ satisfies the equation, confirming the algebraic solution, while other values create false statements Testing by substitution: For $x = 2$ : $2(2) - 3 = 1$ and $2 + 1 = 3$ , so $1 = 3$ (false). For $x = 3$ : $2(3) - 3 = 3$ and $3 + 1 = 4$ , so $3 = 4$ (false). For $x = 4$ : $2(4) - 3 = 5$ and $4 + 1 = 5$ , so $5 = 5$ (true). For $x = 5$ : $2(5) - 3 = 7$ and $5 + 1 = 6$ , so $7 = 6$ (false). Only $x = 4$ works, confirming the algebraic solution.
From the set $\{3,4,5,6\}$ , which values make the inequality $x\ge 5$ true? (Test each value.)
Answer: $\{5,6\}$ Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For the inequality x≥5 with the set {3,4,5,6}, the solving process involves determining which values satisfy it by substituting each one. Substitution: x=5:5≥5 true (equal counts), x=6:6≥5 true, x=3:3≥5 false, x=4:4≥5 false. In contrast, an equation has typically one solution. So solutions are {5,6}. The correct set is identified by substitution, including the boundary for ≥. A common error is excluding 5, thinking ≥ is like >, or not testing all.
From the set $\{0,1,2,3,4\}$ , which values make the inequality $2x<6$ true? (Check each value by substitution.)
Answer: $\{0,1,2\}$ This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality 2x<6 and the test set {0,1,2,3,4}, where you determine which values satisfy it by substitution: pick a value like x=2, replace (2*2<6 becomes 4<6), evaluate and check if true—yes, so x=2 is a solution; inequalities can have multiple solutions, unlike equations which typically have one. For example, with inequality 2x<6 and set {0,1,2,3,4}, check x=0:0<6 true, x=1:2<6 true, x=2:4<6 true, x=3:6<6 false, x=4:8<6 false, so solutions are {0,1,2}. Correct solution identification comes from substituting, multiplying, and checking the inequality. Common errors include including boundary like x=3 when 6<6 is false, arithmetic errors like 2*3=5, or not testing all. Testing systematically: (1) list the set {0,1,2,3,4}, (2) test each like x=3:6<6 false, (3) mark true values like x=0,1,2, (4) collect solutions {0,1,2}. Remember < means strictly less, excluding equality.
A student tests values for the equation $2x=14$ using the set $\{5,6,7,8\}$ . Which value(s) from the set make the equation true?
Answer: $\{7\}$ This question tests understanding of solving equations as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given equation 2x=14 and the test set {5,6,7,8}, where you determine which values satisfy it by substitution: pick a value like x=7, replace (2*7=14), evaluate (14=14), and check if true—yes, so x=7 is a solution; equations typically have one solution. For example, with equation 2x=14 and set {5,6,7,8}, check x=5:10≠14 false, x=6:12≠14 false, x=7:14=14 true, x=8:16≠14 false, so only x=7 is the solution. Correct solution identification comes from substituting, multiplying, and verifying equality. Common errors include wrong values like x=6 when 12≠14, arithmetic like 2*7=12, or claiming multiple solutions. Testing systematically: (1) list the set {5,6,7,8}, (2) test each like x=6:12≠14 false, (3) mark true values like x=7 true, (4) collect solutions {7}. No solution if none match, but here there is one.
Two students are checking if $x = -2$ is a solution to $3(x + 4) = 2x + 8$ . Student A gets: $3(-2 + 4) = 3(2) = 6$ and $2(-2) + 8 = 4$ , so $6 = 4$ (false). Student B gets: $3(-2 + 4) = 3(2) = 6$ and $2(-2) + 8 = -4 + 8 = 4$ , so $6 = 4$ (false). What can you conclude?
Answer: Both students computed correctly and properly concluded that $x = -2$ does not satisfy the given equation Both students performed the substitution correctly. For the left side: $3(-2 + 4) = 3(2) = 6$ . For the right side: $2(-2) + 8 = -4 + 8 = 4$ . So the equation becomes $6 = 4$ , which is false. Therefore $x = -2$ is not a solution. Both students reached the correct conclusion through correct arithmetic. Student A's work shown is incomplete but the final answer is right.
A student is checking solutions by substitution. Which values from the set $\{4,5,6,7\}$ make the equation $x+8=14$ true?
Answer: $\{6\}$ This question tests understanding of solving equations as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given equation x + 8 = 14 and the test set {4,5,6,7}, where you determine which values satisfy it by substituting each one. For example, try x=6: replace the variable to get 6 + 8 = 14, evaluate to 14 = 14, which is true, so x=6 is a solution; equations like this typically have one solution, unlike inequalities which can have multiple. For instance, checking the set: x=4 gives 4+8=12≠14 false, x=5 gives 5+8=13≠14 false, x=6 gives 6+8=14=14 true, x=7 gives 7+8=15≠14 false, so only x=6 is the solution. The correct solution is identified by substitution as {6}. A common error is an arithmetic mistake, like claiming x=5 works because 5+8=14 incorrectly, or not testing all values and missing the solution. To test systematically: (1) list the set {4,5,6,7}, (2) substitute each into x+8=14 and evaluate, (3) mark true values like x=6, (4) collect solutions as {6}; remember, if no value works, report no solution in the set.
Marcus is checking which values from the set { $-2, -1, 0, 1, 2$ } make the inequality $3x + 4 > x - 2$ true. He substitutes each value and gets these results: For $x = -2$ : $-2 > -6$ (true), For $x = -1$ : $1 > -3$ (true), For $x = 0$ : $4 > -2$ (true), For $x = 1$ : $7 > -1$ (true), For $x = 2$ : $10 > 0$ (true). What error did Marcus make?
Answer: He evaluated $x - 2$ incorrectly, writing $-6$ instead of $-4$ when $x = -2$ When $x = -2$ , Marcus should get $x - 2 = -2 - 2 = -4$ , not $-6$ . The correct comparison should be $-2 > -4$ (true). Marcus made an arithmetic error. Choice A is wrong because he substituted correctly into both sides. Choice B is wrong because $3(-2) + 4 = -2$ and $3(-1) + 4 = 1$ are correct. Choice C is wrong because substitution is the correct method for this problem.
A student is checking which locker code works. From the set $\{5,6,7,8\}$ , which value of $x$ makes the equation $x+5=12$ true? (Test by substitution.)
Answer: $x=7$ Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For the equation x+5=12 with the set {5,6,7,8}, the solving process involves determining which values satisfy it by substituting each one. For example, try x=7: replace x in x+5=12 to get 7+5=12, evaluate to 12=12, which is true, so x=7 is a solution; whereas for x=5: 5+5=10≠12, false. In contrast, an inequality like x>4 from {2,3,4,5,6} would have multiple solutions such as {5,6} after testing each. For this equation x+5=12 with set {5,6,7,8}, check x=5: 5+5=10≠12 false, x=6:6+5=11≠12 false, x=7:7+5=12=12 true, x=8:8+5=13≠12 false, so only x=7 is the solution. The correct solution is identified by substitution, and here it's x=7. A common error is an arithmetic mistake, like claiming 7+5=11, or picking a value not in the set.
From the set { $-1, 0, 1, 2$ }, which value(s) make the inequality $x^2 + 1 ≤ 2$ true when substituted?
Answer: Values $x = -1, 0,$ and $1$ all satisfy the inequality when substituted and computed correctly Substituting each value into $x^2 + 1 ≤ 2$ : For $x = -1$ : $(-1)^2 + 1 = 1 + 1 = 2$ , and $2 ≤ 2$ is true. For $x = 0$ : $0^2 + 1 = 1$ , and $1 ≤ 2$ is true. For $x = 1$ : $1^2 + 1 = 2$ , and $2 ≤ 2$ is true. For $x = 2$ : $2^2 + 1 = 5$ , and $5 ≤ 2$ is false. So $x = -1, 0, 1$ all work.
From the set $\{2,3,4,5\}$ , which values make the inequality $x<4$ true? (Test each value.)
Answer: $\{2,3\}$ This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality x<4 and the test set {2,3,4,5}, where you determine which values satisfy it by substitution: pick a value like x=3, replace (3<4), evaluate and check if true—yes, so x=3 is a solution; inequalities can have multiple solutions from the set. For example, with inequality x<4 and set {2,3,4,5}, check x=2:2<4 true, x=3:3<4 true, x=4:4<4 false, x=5:5<4 false, so solutions are {2,3}. Correct solution identification comes from testing each and confirming strict inequality. Common errors include including x=4 when < excludes equality, or not testing all values. Testing systematically: (1) list the set {2,3,4,5}, (2) test each like x=4:4<4 false, (3) mark true values like x=2 and x=3, (4) collect solutions {2,3}. Remember < means less than, not less than or equal.

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